This may not at first seem obvious, but if you have calculated the th Taylor Polynomial, you can find the th Taylor polynomial by adding one more term. That is

Some Things to Notice

The 1st Taylor polynomial centered at (respectively ) is just the usual linear approximation that you did in first semester calculus, since (respectively ). I.e., this is the tangent line to the function at the point (respectively ).

As your Taylor polynomials become higher and higher in degree, they approximate the function “locally” (i.e., near or near ) better and better. This is because you are encoding information about the value of the function at or , then the value of the derivative, then the value of the second derivative and so on.

One of the important things to realize is that a higher degree Taylor polynomial will represent the function better for values closer to the center of the Taylor polynomial. If you graph this situation for higher and higher degree polynomials, this makes more sense. I am placing a few graphs on the website that you can examine to see what I mean here.

The Taylor polynomial will be identical to the function at the point where the Taylor polynomial is centered. This is easy to see, since all of the components (those other than the first term of the Taylor polynomial) are zero. Also, the th Taylor polynomial of an th degree polynomial will be identical to the polynomial itself. I'll do an example of this below.

You can only create Taylor polynomials when the value of the function at and the value of the function's derivatives at can be determined.

Examples

Find the first through fifth Taylor polynomials of centered at .

Solution: Since

and all higher derivatives are zero, we find the following.

You should notice that this first Taylor polynomial is just the tangent line approximation to .

Since the second derivative is zero, the first and second Taylor polynomials are identical.

The main point here is that . Also, since the remaining derivatives are zero, we have

Find the 3rd Taylor polynomials of , and around . Now calculate the same polynomials around .

Solution: Your text does this quite well for the polynomials centered around , so I'll put the cases here. In many cases, you can simply write out the polynomial term by term (skipping entirely any terms with a derivative equal to zero) if all of the derivatives are easy to calculate, which they are in this situation.

rd Degree Taylor Polynomials Centered at

Notice that these have some considerable difference with the usual Taylor polynomials centered at .

Find the th degree Taylor polynomial for around .

Solution: This is an interesting Taylor polynomial, since the function is simply the function shifted to the left by one unit and isn't defined for . So, we know that is not defined for . The key idea here is that the Taylor polynomials are nice because they are polynomials (and polynomials are easy to work with), but they may not accurately reflect the functions they approximate as you move farther away from where the Taylor polynomials are centered (in this case, the center is zero). Thus, we expect out Taylor polynomials to have some value at, say , while isn't defined there. The derivatives we will need are as follows.

Now, I'll admit that this next step takes practice. What we want to do is to generalize any pattern that we see coming out of these derivatives. Notice that every other derivative is negative. If we then consider the absolute values, we have the sequence which is the factorial sequence. (Do more derivatives if you need convincing of this.) So, we have the th derivative given by

Remember that the formula for forming a Taylor polynomial includes dividing by factorials for each term. Therefore, the th Taylor polynomial for around is given by

Taylor Polynomials You Should Know

Some of this will be repeated a bit in coming sections, but I will record it here for completeness. These are all centered at . If you want the th Taylor polynomial, you just consider any of the following expansions until you reach or surpass in degree.